On locally extremal functions on connected spaces

ON LOCALL Y EXTREMAL FUNCTIONS ON CONNE CTED SP A CES T. BANAKH, M. VO VK, M. W ´ OJCIK Abstract. W e construct an exa mple of a con tin uous non-constan t function f : X → R defined on a connect ed complete metric space X suc h that ev ery point x ∈ X i s a p oint of lo cal mini mum or lo cal maximum for f . Also w e sho w that an y s uc h a space X s hould hav e the netw ork weigh t and the weak separate num ber nw( X ) ≥ R ( X ) ≥ c . This note is motiv a ted by the following problem of M.W´ ojcik [W´ oj], [MW] (see also [BGN, Question 7]) on the nature of loca lly extremal fun ctions on co nnec ted spaces. W e define a function f : X → R on a top olog ical spa ce X to b e lo c al ly extr emal if each p oint x ∈ X is a p oint of lo cal minimum or lo cal max imum for f . Problem 1. Is a e ach lo c al ly ext r emal c ontinuous funct ion f : X → R on a c onne cte d m etric sp ac e X c onstant? In [BGN] this problem has b een answered in affirmative under the condition that the density of the connected metric space X is str ictly smaller than c , the size of con tin uum. In fact, this result is a particular c a se a more genera l theorem treating loc ally extremal functions on connected topologica l spaces X with the w eak separate num ber R ( X ) < c . F ollowing M.Tk achenk o [Tk], we define a top olog ical space X to b e we akly sep- ar a te d if to ea ch p oint x ∈ X has an op en neig hborho o d O x ⊂ X such that for any t wo distinct p oints x, y ∈ X e ither x / ∈ O y or y / ∈ O x . The cardina l R ( X ) = sup {| Y | : Y is a weakly separated subspac e of X } is c a lled t he we ak sep ar ate numb er o f X . By [Tk], R ( X ) ≤ nw ( X ) ≤ w ( X ), wher e w ( X ) (resp. nw ( X )) stands fo r the (netw ork) weigh t of X . On the o ther hand, A. Ha jnal and I. Juhas z [HJ] constructed a CH-exa mple o f a regular space X with ℵ 0 = R ( X ) < nw ( X ) = c . It is a n op en pro blem if such a n example e xists in Z FC, see Pr oblem 15 in [GM]. Theorem 1. If X is a top olo gic al sp ac e a nd f : X → R is a lo c al ly extr emal function, then | f ( X ) | ≤ R ( X ) . Pr o of. W rite X as the union X = X 0 ∪ X 1 of the sets X 0 and X 1 consisting of lo cal minim ums and lo cal maximum s of the function f , resp ectively . W e cla im that | f ( X 0 ) | ≤ R ( X ). Assuming the converse, find a subset A ⊂ X 0 such that | A | > R ( X ) a nd f | A is injectiv e. Each po int a ∈ A , b eing a point of lo cal minim um of f , p oss esses a neigh bo rho o d O a ⊂ X such that f ( a ) ≤ f ( x ) for all x ∈ O a . W e claim that the family o f neighborho o ds { O a } a ∈ A witnesses that the set A is w eakly separated. Assuming the opp osite, we would find tw o distinct po int s a, b ∈ A such 1991 Mathematics Subje ct Classific ation. 54D05; 54C30. 1 2 T. BANAKH, M. VO VK, M. W ´ OJCIK that a ∈ O b and b ∈ O a . It follows from b ∈ O a that f ( a ) ≤ f ( b ) a nd from a ∈ O b that f ( b ) ≤ f ( a ). Consequently , f ( a ) = f ( b ), whic h con tradicts the injectivity of f on A . This contradiction proved the inequality | f ( X 0 ) | ≤ R ( X ). By analog y we can prove that | f ( X 1 ) | ≤ R ( X ). If R ( X ) is infinite then the inequality ma x {| f ( X 0 ) | , | f ( X 1 ) |} ≤ R ( X ) implies | f ( X ) | ≤ | f ( X 0 ) | + | f ( X 1 ) | ≤ 2 R ( X ) = R ( X ). Now a ssume that n = R ( X ) is finite. W e claim that | f ( X ) | ≤ n . Assuming the converse, we would find a finite subset A = { a 0 , . . . , a n } ⊂ X such that | f ( A ) | = n + 1 . The injectivity and co n tinuit y o f the map f | A guar antees that the s ubs pace A is discr e te and hence weakly separated. Consequently , R ( X ) ≥ | A | = n + 1 > R ( X ), which is a contradiction.  Corollary 1. Each c ontinuous lo c al ly extr emal function f : X → R on a c onn e cte d top olo gic al sp ac e X with R ( X ) < c is c onstant. Pr o of. By Theo r em 1, f ( X ) is a connected subset of R with cardinality | f ( X ) | ≤ R ( X ) < c , which is p o ssible o nly if f ( X ) is a singleton.  The condition R ( X ) < c is esse ntial in this corollar y a s the following ex a mple from [MW] and [BGN] shows. Example 1. The pr oje ction f : [0 , 1 ] 2 → [0 , 1] of the lexic o gr aphic squar e ont o the interval is lo c ally extr emal but not c onstant . In [DF] A. Le Donne and A. F edeli announced the existence of a non-cons tant lo cally extremal con tin uous function defined o n a connec ted metric space. In the following theorem we describ e a completely-metriza ble example with the same pro p- erties. Theorem 2. Ther e is a c onn e cte d c ompl ete m etric sp ac e and a c ontinuous fun ction f : X → R which is lo c al ly ex tr emal but not c onstant. Pr o of. Let I ↑ = { x ↑ : x ∈ I } and I ↓ = { x ↓ : x ∈ I } b e tw o disjoint copies of the unit int erv al I = [0 , 1]. A basic building blo ck of our metric spac e X is the full g raph T = [ x,y ∈ I ↓ ∪ I ↑ [ x, y ] ⊂ l 1 ( I ↓ ∪ I ↑ ) with the set of vertices I ↓ ∪ I ↑ in the Banach space l 1 ( I ↓ ∪ I ↑ ) = { g : I ↓ ∪ I ↑ → R : X x ∈ I ↓ ∪ I ↑ | f ( x ) | < ∞} . F or every α ∈ [0 , 1 ] the co py T α = T × { α } of T w ill be ca lled the α th town. The space T α has diameter 2 in the metric d α induced from l 1 -metric of the Banach space l 1 ( I ↓ ∪ I ↑ ). It will be con venien t to think of the distance in the town T α as the smallest amount of time for g e tting from one pla ce to another place of T α by a taxi that mov es with velo city 1 . The vertices x ↓ α = ( x ↓ , α ) ∈ I ↓ × { α } ⊂ T α , x ↑ α = ( x ↑ , α ) ∈ I ↑ × { α } ⊂ T α of the graph T α are ca lle d low er and upper airp orts, r e s p e c tively . F or a ny indices γ < α < β in I , b etw een the airp o rts α ↑ α ∈ T α and α ↓ β ∈ T β there is an air connection taking β − α units o f time. Similarly , b etw een the a irp orts α ↓ α and α ↑ γ there is an air connection taking α − γ units of time. ON LOCALL Y E XTREMAL FUNCTIONS ON CONNECTED SP ACES 3 Now define a metric d on the space X = S α ∈ I T α as the sma lle st amount of time necess ary to g e t from one place to another plac e of X using taxi (inside of the towns) a nd planes (betw een the towns). More formally , this metric d o n X can b e defined as follows. In the squar e X × X consider the subset D = [ α ∈ I T α × T α ∪ [ α<β { ( α + α , α − β ) , ( α − β , α + α ) , ( β − β , β + α ) , ( β + α , β − β ) } and define a function ρ : D → R letting ρ ( x, y ) = ( d α ( x, y ) if x, y ∈ T α , α ∈ I ; | β − α | if { x, y } ∈  { α + α , α − β } , { β − β , β + α }  for so me α < β in I . This function induces a metric d on X defined by d ( x, y ) = inf n n X i =1 ρ ( x i − 1 , x i ) : ∀ i ≤ n ( x i − 1 , x i ) ∈ D, x 0 = x, x n = y o . It is easy to chec k that d is a complete metric o n X . Next, define a ma p f : X → I ⊂ R letting f − 1 ( α ) = T α for α ∈ I . It is easy to see that this map is non-consta nt and non-e x panding and hence c ontin uous. Claim 1. Each p oint x ∈ X is a p oint of lo c al minimum or lo c al maximum of the map f : X → I . Pr o of. T ake a ny p oint x ∈ T α ⊂ X , α ∈ I . If x / ∈ I ↓ ∪ I ↑ is not an air p ort, then we can find ε > 0 such that the o p en ε -ball O ε ( x ) of x in T α do es not intersects the set I ↓ α ∪ I ↑ α . In this cas e O ε ( x ) is op en in X and hence f is lo cally constant at x . Now assume that x ∈ I ↓ α ∪ I ↑ α and hence x = β ↓ α or β ↑ α for some β ∈ I . First consider the case when x = β ↓ α coincides with a low er airp o rt. If β = α , then we can consider the 1- neighborho o d ball O 1 ( x ) aro und the airp ort x = α ↓ α and observe that this neig hborho o d do es no t intersect towns T γ with γ > α . Consequently , f ( O 1 ( x )) ⊂ ( −∞ , α ] = ( − ∞ , f ( x )], which mea ns that x is a p oint of lo cal maximum of f . If β 6 = α , then we consider the op en ball O ε ( x ) ⊂ X of r adius ε = β − α centered at x = β ↓ α and o bserve that it contains o nly p oints of the town T α , which means that f is lo cally c o nstant at x . The ca se x = I ↑ α can b e considered by a nalogy .  Claim 2. The metric sp ac e X is c onne ct e d. Pr o of. Given a non- empt y op en-a nd-closed subset U ⊂ X , we should pr ov e that U = X . First w e sho w that the image f ( U ) is o p e n. Given an y p o in t α ∈ f ( U ), fix any x ∈ U ∩ T α . It follows from the connectedness of the town T α that T α ⊂ U . Consider the airpo rts α ↓ α , α ↑ α and find ε > 0 such that the op en set U c ontains the op en ε -ba lls O ε ( α ↓ α ) and O ε ( α ↑ α ) cen tered at those airp or ts. It follo ws fro m the definition o f the metr ic d on X that f ( O ε ( α ↓ α )) ⊃ ( α − ε, α ] ∩ I and f ( O ε ( α ↑ α )) ⊃ [ α, α + ε ) ∩ I . Unifying those inclusions, we get f ( U ) ⊃ ( α − ε, α + ε ) ∩ I , witness ing that f ( U ) is op en in I . By the same reaso n ( X \ U ) is op en in I . The connectedness of the fib ers T α of f implies that the sets f ( U ) and f ( X \ U ) a re disjoint. Now the co nnectedness of the interv al I = f ( U ) ∪ f ( X \ U ) guara ntees that f ( X \ U ) is empt y and hence U = X .  4 T. BANAKH, M. VO VK, M. W ´ OJCIK  The space X fr o m Theorem 2 has an interesting pr o p erty: it is connected but no t separably co nnected (and hence not pa th connected). F ollowing [CHI] or [BEHV], we define a top olog ical spa ce X to b e sep ar ably c onne cte d if any t wo p oints x, y ∈ X lie is a connected separable subspace of X . The pr oblem o n the existence of connected metric spaces that ar e not separ ably connected w as pos ed in [BEHV] and was a ns wered in [AM] and [WPhD]. How ev er all known examples of such spaces are nor c omplete. Theorem 3. Ther e is a c omplete metric sp ac e which is c onne cte d but not s ep ar ably c onne cte d. Pr o of. The connected complete metric space X fr om Theorem 2 is not separ ably connected by Corolla r y 1.  The connected metric space X fro m Theo rem 2 is not s eparably connected but contains many non-dege ne r ate separa ble connected subspaces. Problem 2. Is ther e a c onne cte d c omplete metric sp ac e C such that e ach c onne cte d sep ar able su bsp ac e of C is a singleton. Non-complete connected metric spa c e s with this prop erty w ere co ns tructed in Theorem 28 of [WP hD]. ON LOCALL Y E XTREMAL FUNCTIONS ON CONNECTED SP ACES 5 References [AM] R. A ron, M. 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T.Banakh: Iv an Franko L viv Na tional University, L viv, Ukraine, an d Unwersytet Hu m anistyczno-Przyrodnicz y im. Jana Kochanowskiego, Kielce, Poland E-mail addr ess : tbanakh@yah oo.com M.Vo vk: Na tional University “L vivska Politechnika”, L viv , Ukraine M.R.W ´ ojcik: Instytut Ma tema tyki i Informa tyki, Politechnika Wrocla wska, Wro- cla w, Poland E-mail addr ess : michal.r.wo jcik@pwr.w roc.pl